Pre-Renaissance painting struggled to portray a three dimensional world on a two dimensional canvas. Some examples include
http://commons.wikimedia.org/wiki/Image:Altichiero_da_Zevio_001.jpg
http://www.mos.org/sln/Leonardo/PaintingBeforetheRen.html
All of these paintings use some kind of occlusion or painting of one object partial blocking another in an attempt to convey a sense of depth. Instead of using multiple depth cues, such as linear perspective, aerial perspective, shading and size principles, or texture gradient, these artists relied almost entirely on occlusion. For small illusions, occlusion is an acceptable, albeit unsophisticated, style of portraying depth, but by no means was it the most effective. It was not until the Renaissance that artists started to incorporate these alternative depth cues into their paintings along with occlusion, and it is from these paintings that the modern principles of depth painting emerged. A good painter could manipulate the addition of these depth cues to convince the brain that it is in fact observing a three dimensional world, while it was merely observing a flat piece of canvas. The primary development during the Renaissance was the mastering of shading techniques, which provide a source of light and can really provide depth, texture, and feeling in an artistic work, especially in folds of cloth.
Artists can also use color and linear perspective very effectively to create depth. Bright, vibrant colors in the foreground followed by medium tones in the middle and cooler, less saturated colors in the background really make you think the scene is real and three dimensional. Linear perspective is very often used, especially in landscape scenes where lines can run all the way to the horizon and converge at the vanishing point. This provides a fantastic sense of depth and really lengthens the image.
You can see examples of color and vanishing point at work in these images:
http://www.mydigitalnoise.com/images/20051031212952_vanishing%20point%20-%20small.jpg
http://www.la-art-tutor.com/wp-content/uploads/2007/10/oppd11.jpg
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